Extended 1D Inversion Of Electromagnetic Measurements For Formation Evaluation

ABSTRACT

A method to determine at least one formation property of a subterranean formation includes providing a downhole electromagnetic logging tool having at least one transmitter array and one receiver array and acquiring measurements in the formation using the transmitter and receiver arrays of the downhole electromagnetic logging tool. The method further includes performing a first inversion in response to the measurements, wherein at least one of dip or dip azimuth are assumed constant in an inversion zone within the formation to obtain an inverted formation model that comprises at least one of horizontal resistivity (Rh), vertical resistivity (Rv), dip, and dip azimuth. The method includes determining an n-th order partial derivative matrix of at least one of dip or dip azimuth, wherein n is greater than or equal to 1. The method includes performing a second inversion using the determined n-th order partial derivative matrix, wherein at least one of dip and dip azimuth are allowed to vary in the inversion zone, to obtain an updated formation model. The method includes determining the at least one formation property of the formation using the up-dated formation model.

BACKGROUND

1. Technical Field

The present disclosure relates generally to the field of electromagneticwell logging techniques. More specifically, the present disclosurerelates to providing improved inversion techniques for determiningcharacteristics of a subsurface formation based on electromagneticmeasurements obtained using a well logging tool disposed in a borehole.

2. Background Information

This section is intended to introduce the reader to various aspects ofart that may be related to various aspects of the subject matterdescribed and/or claimed below. This discussion is believed to behelpful in providing the reader with background information tofacilitate a better understanding of the various aspects of the presentdisclosure. Accordingly, it should be understood that these statementsare to be read in this light, not as admissions of prior art.

Logging tools have long been used in wellbores to make, for example,formation evaluation measurements to infer properties of the formationssurrounding the borehole and the fluids in the formations. As examplesonly, common logging tools include electromagnetic tools, nuclear tools,and nuclear magnetic resonance (NMR) tools. Electromagnetic loggingtools typically measure the resistivity (or its reciprocal,conductivity) of a formation. For instance, such electromagneticresistivity tools include galvanic tools, induction tools, andpropagation tools. Typically, a measurement of the attenuation and phaseshift of an electromagnetic signal that has passed through the formationis used to determine the resistivity and/or other characteristics of theformation.

The concept of multiaxial (e.g., triaxial) electromagnetic measurementscan be traced back to the 1960s, with the concept having been describedin U.S. Pat. No. 3,014,177 to Hungerford. Since then, various triaxialelectromagnetic tools have been introduced. See, e.g., Kriegshäuser etal., “A New Multicomponent Induction Logging Tool to Resolve AnisotropicFormations,” Transactions of the SPWLA 41st Annual Logging Symposium,Paper D (2000). For instance, triaxial electromagnetic tools may includea multiarray electromagnetic induction tool having collocatedtransmitter and/or receiver coils, such as of a type described generallyin Barber et al., “Determining Formation Resistivity In The Presence ofInvasion,” SPE 90526, presented at the SPE Annual Technical Conferenceand Exhibition, Houston, Tex., Sep. 26-29, 2004, or in Rosthal et al.,“Field Test Results of an Experimental Fully-Triaxial Induction Tool,”Transactions of the SPWLA 44th Annual Logging Symposium, Paper QQ(2003). Additional multi-array triaxial induction tools continue to bedeveloped. See, e.g., Hou et al., WO Patent Publication No.WO2011/123379A1; see also Hou et al., “Real-time Borehole Correction Fora New Multicomponent Array Induction Logging Tool in OBM Wells,”Transactions of the SWPLA 42th Annual Logging Symposium, Paper PPP(2012).

When compared to certain older conventional tools, some of the morerecent electromagnetic tools include those that incorporate mutuallyorthogonal coils and/or tilted coils to the tool design, which can serveas a transmitter, a receiver, or both (e.g., a transceiver). Theincorporation of such coils with off-borehole axis orientation enablesmeasuring a full, a nearly full, or a partially full tensor measurementinstead of, for example, just a zz-component, as would be the case insome conventional tools with off-borehole axis coil arrangements. Suchan expanded set of measurements may allow for extraction of resistivityanisotropy and geometry of subsurface formations. For example,multiaxial electromagnetic logging instruments can also be used todetermine the relative dip angle (θ) and dip azimuth angle (Φ) of rockformations as well as anisotropic formation resistivities, includingvertical resistivity (Rv) and horizontal resistivity (Rh). The apparentconductivity tensor measured by triaxial electromagnetic tools issensitive to these formation parameters.

Several methods are known for solving for such formation parameters,each having certain advantages and disadvantages. For instance, onecategory of these methods is based on a formation model where bedboundaries are neglected, and a uniform anisotropic formation model isassumed. This is sometimes referred to as a 0D (zero-D) inversion or a0D formation model. Example implementations of 0D models are describedin more detail in commonly owned International Patent ApplicationPublication No. WO2011/091216, which is hereby incorporated by referencein its entirety. Some advantages of 0D inversions are that they arecomputationally fast and thus are often suitable for use as a real-timeanswer product for generating coarse estimates of formation properties,and that they generally provide high-resolution dip estimation. 0Dinversions can provide good results when used in formations with weakresistivity contrast and/or a slow varying resistivity. However, whenthe resistivity contrast in a given formation is large, 0D inversionalgorithms may be negatively impacted by shoulder bed effects (e.g., bedboundary effects), as 0D formations models can be considered“simplified” in the sense that they do not include bed boundaries in theformation model. This can result in inaccuracies and adverse effects onboth resistivity and dip estimations.

To overcome the shoulder bed effect, bed boundaries may be incorporatedin the formation model. The simplest formation model that accounts forshoulder bed effects is a 1D formation model (e.g., based on vertical 1Dalgorithm). The basic assumption of a 1D formation model is that dip andazimuth are constant across the whole model, although resistivity (Rhand Rv) may vary from bed to bed. In other words, for a 1D model, eachbed is assigned a different anisotropy characterized by horizontalresistivity (Rh) and vertical resistivity (Rv), with all the bedboundaries are assumed to be parallel to each other. The normal to thebedding plane is allowed to be tilted relative to the borehole axis.Accordingly, the resultant orientation is characterized by formation dipand dip azimuth.

These parameters can be recovered with a 1D inversion for petrophysicaland geological applications. When applied to actual data, thecorresponding depth zone is segmented first into a number of smallsubzones for the ease of 1D inversion implementation. 1D inversion isrun over all subzones sequentially first, and the inversion results fromall subzones are then combined as the final results for the whole depthzone. Each of these subzones is referred to as an inversion zone, or aninversion window. For example, an inversion zone with a length of up to100ft has been reported to be used for some 1D inversion methods. It isalso noted that 1D inversions are also relatively fast from acomputational standpoint and thus can be used for real-time ornear/substantially real-time applications (providing of results withoutan appreciable delay). Thus, from an industry perspective, 1D inversiontechniques are generally considered as a suitable tradeoff between themodel fitness (e.g., compared to 0D inversion techniques) andcomputational cost. Examples of 1D inversion techniques are described inWang et al., “Triaxial Induction Logging: Theory, Modeling, Inversion,and Interpretation,” SPE 103897, SPE International Oil & Gas Conferenceand Exhibition, Dec. 5-7, 2006.

However, as the use of triaxial electromagnetic logging has increased inpopularity, field data sets have shown that formations are often not 1Din the real world. Rather, variations in dip and azimuth are oftenobserved over the length of a given inversion zone. Such variationsviolate the assumptions of a 1D formation model, and thus indicate thatin principle, 1D formation models may not be the best choice formodeling these so-called complex formations. Further, in extreme cases,variation of dip and dip azimuth in the formation may lead to largemodel errors in the data, which may translate to large errors inresistivity calculations and dip estimates obtained using 1D inversions.

One way of overcoming the drawbacks of 1D inversions in complexformations is to use even higher-dimension inversions, such as 3Dinversions which are capable of modeling earth formations even moreaccurately. As will be appreciated, 3D inversions have been applied inprocessing cross-well electromagnetic measurements and for controlledsource electromagnetic measurements in marine environments. In theory,higher-dimension inversions, such as a 3D inversion, often provide moreaccurate data by reducing or eliminating model errors. However, from apractical standpoint, such methods are generally time-consuming andrequire vast computing resources and processing capabilities whichgenerally render them unsuitable for providing real-time or nearreal-time data and answer products. Furthermore, the inevitable increasein the number of unknowns makes the inversion overall more difficult dueto non-uniqueness issues (e.g., can result in multiple models that fitthe data).

SUMMARY

A summary of certain embodiments disclosed herein is set forth below. Itshould be understood that these aspects are presented merely to providethe reader with a brief summary of these certain embodiments and thatthese aspects are not intended to limit the scope of this disclosure.Indeed, this disclosure may encompass a variety of aspects that may notbe set forth below.

In one illustrative embodiment, a method to determine at least oneformation property of a subterranean formation includes providing adownhole electromagnetic logging tool having at least one transmitterarray and one receiver array and acquiring measurements in the formationusing the transmitter and receiver arrays of the downholeelectromagnetic logging tool. The method further includes performing afirst inversion in response to the measurements, wherein at least one ofdip or dip azimuth are assumed constant in an inversion zone within theformation to obtain an inverted formation model that comprises at leastone of horizontal resistivity (Rh), vertical resistivity (Rv), dip, anddip azimuth. The method further includes determining an n-th orderpartial derivative matrix of at least one of dip or dip azimuth, whereinn is greater than or equal to 1. The method further includes performinga second inversion using the determined n-th order partial derivativematrix, wherein at least one of dip and dip azimuth are allowed to varyin the inversion zone, to obtain an updated formation model.Additionally, the method includes determining the at least one formationproperty of the formation using the updated formation model.

In a further illustrative embodiment, a system includes anelectromagnetic logging tool having at least one transmitter array andone receiver array, the electromagnetic logging tool being configured toacquire measurements in a subterranean formation using the transmitterand receiver arrays when disposed in the formation. The system alsoincludes a processing device that performs a first inversion in responseto the measurements wherein at least one of dip or dip azimuth areassumed constant in an inversion zone within the formation to obtain aninverted formation model that comprises at least one of horizontalresistivity (Rh), vertical resistivity (Rv), dip, and dip azimuth,determines an n-th order partial derivative matrix of at least one ofdip or dip azimuth, wherein n is greater than or equal to 1, performs asecond inversion using the determined n-th order partial derivativematrix, wherein at least one of dip and dip azimuth are allowed to varyin the inversion zone, to obtain an updated formation model, anddetermines at least one formation property of the formation using theupdated formation model.

In yet another illustrative embodiment, a non-transitorycomputer-readable medium having computer executable instructions forcausing a computer to, in response to electromagnetic measurementobtained by an electromagnetic logging tool disposed in a formation,perform a first inversion wherein at least one of dip or dip azimuth areassumed constant in an inversion zone within the formation to obtain aninverted formation model that comprises at least one of horizontalresistivity (Rh), vertical resistivity (Rv), dip, and dip azimuth,determine an n-th order partial derivative matrix of at least one of dipor dip azimuth, wherein n is greater than or equal to 1, perform asecond inversion using the determined n-th order partial derivativematrix, wherein at least one of dip and dip azimuth are allowed to varyin the inversion zone, to obtain an updated formation model, anddetermine at least one formation property of the formation using theupdated formation model.

Again, the brief summary presented above is intended only to familiarizethe reader with certain aspects and contexts of embodiments of thepresent disclosure without limitation to the claimed subject matter.

BRIEF DESCRIPTION OF THE DRAWINGS

Various aspects of this disclosure may be better understood upon readingthe following detailed description and upon reference to the drawings inwhich:

FIG. 1A is a schematic diagram of a subterranean wireline well loggingsystem that includes an electromagnetic logging tool, wherein the systemof FIG. 1 is capable of performing an inversion on electromagneticmeasurements acquired using the electromagnetic logging tool to obtainformation evaluation parameters in accordance with aspects of thepresent disclosure;

FIG. 1B is a schematic diagram of a subterranean logging-while-drillingand/or measure-while-drilling well logging system that includes anelectromagnetic logging tool, wherein the system of FIG. 1B is capableof performing an inversion on electromagnetic measurements acquiredusing the electromagnetic logging tool to obtain formation evaluationparameters in accordance with aspects of the present disclosure;

FIG. 2A illustrates an example triaxial transmitter and receiverarrangement in accordance with aspects of the present disclosure;

FIG. 2B illustrates a multiaxial resistivity tensor that may be measuredusing the transmitter and receiver arrangement of FIG. 2A;

FIG. 2C illustrates an example of a transverse transmitter and tiltedreceiver arrangement in accordance with aspects of the presentdisclosure;

FIG. 3 shows graphs of horizontal resistivity, vertical resistivity, dipand dip azimuth over a given depth zone, and also shows a graphdepicting the relative error between a 1D model (using a zero-th orderTaylor series expansion for dip and dip azimuth) and an exact 3D modelfor various array spacings of an electromagnetic logging tool inaccordance with aspects of the present disclosure;

FIG. 4 shows graphs depicting relative error between severalapproximated 3D models using Taylor series expansion and exact 3D modelsfor various array spacings of an electromagnetic logging tool inaccordance with aspects of the present disclosure;

FIG. 5 is a graph showing the reduction of data misfit metrics incomparison to a 1D formation model for several different 3D modelapproximations using first and second order Taylor series expansionapproximations in accordance with aspects of the present disclosure;

FIG. 6 is a flowchart depicting a process of performing an extended 1Dinversion in which dip azimuth is an unknown parameter in accordancewith aspects of the present disclosure;

FIG. 7 is a flowchart depicting a process of performing an extended 1Dinversion in which dip azimuth is determined prior to the inversion andprovided to the inversion as a known parameter in accordance withaspects of the present disclosure;

FIG. 8 shows an example of an extended 1D inversion in a 1D formationwith constant dip and dip azimuth in accordance with aspects of thepresent disclosure;

FIG. 9 shows an example of an extended 1D inversion in a non-1D (e.g.,3D) formation with varying dip and dip azimuth in accordance withaspects of the present disclosure;

FIG. 10 shows an example of an extended 1D inversion where dip is notallowed to vary in the inversion zone in accordance with aspects of thepresent disclosure; and

FIG. 11 are example logs from a water-base mud field example furtherillustrating an extended 1D inversion in accordance with aspects of thepresent disclosure.

DETAILED DESCRIPTION OF SPECIFIC EMBODIMENTS

One or more specific embodiments of the present disclosure are describedbelow. These embodiments are only examples of the presently disclosedtechniques. Additionally, in an effort to provide a concise descriptionof these embodiments, all features of an actual implementation may notbe described in the specification. It should be appreciated that in thedevelopment of any such implementation, as in any engineering or designproject, numerous implementation-specific decisions are made to achievethe developers' specific goals, such as compliance with system-relatedand business-related constraints, which may vary from one implementationto another. Moreover, it should be appreciated that such developmentefforts might be complex and time consuming, but would nevertheless be aroutine undertaking of design, fabrication, and manufacture for those ofordinary skill having the benefit of this disclosure.

When introducing elements of various embodiments of the presentdisclosure, the articles “a,” “an,” and “the” are intended to mean thatthere are one or more of the elements. The terms “comprising,”“including,” and “having” are intended to be inclusive and mean thatthere may be additional elements other than the listed elements. Theembodiments discussed below are intended to be examples that areillustrative in nature and should not be construed to mean that thespecific embodiments described herein are necessarily preferential innature. Additionally, it should be understood that references to “oneembodiment” or “an embodiment” within the present disclosure are not tobe interpreted as excluding the existence of additional embodiments thatalso incorporate the recited features.

To provide some general background with respect to the field of welllogging and formation evaluation, FIGS. 1A and 1B illustrate differenttypes of well site systems, which can be deployed onshore or offshore.Specifically, FIG. 1A illustrates a wireline system for investigatingearth formations. As shown, the system includes an electromagneticlogging device 30 for investigating earth formations 31 traversed by aborehole 32. The electromagnetic logging device 30 is suspended in theborehole 32 on an armored cable 33 (e.g., a wireline cable), the lengthof which substantially determines the relative axial depth of the device30. As can be appreciated, the cable length is controlled by suitablemeans at the surface such as a drum and winch mechanism 6. Surfaceequipment 7 can be of conventional types and can include aprocessor-based system which communicates with downhole equipmentincluding electromagnetic logging device 30. The electromagnetic loggingdevice 30 may be a multi-axial electromagnetic logging tool, such as atriaxial tool.

FIG. 1B shows another example of another type of well site system forlogging-while-drilling (LWD) and/or measurement-while-drilling (MWD)applications. Here, a borehole 11 is formed in subsurface formations byrotary drilling in a manner that is well known. Some embodiments canalso use directional drilling. As shown, a drill string 12 is suspendedwithin the borehole 11 and has a bottom hole assembly (BHA) 100 whichincludes a drill bit 105 at its lower end. The surface system includesplatform and derrick assembly 10 positioned over the borehole 11, withthe assembly 10 including a rotary table 16, kelly 17, hook 18 androtary swivel 19. In operation, the drill string 12 is rotated by therotary table 16, energized by means not shown, which engages the kelly17 at the upper end of the drill string. The drill string 12 issuspended from a hook 18, attached to a traveling block (also notshown), through the kelly 17 and a rotary swivel 19 which permitsrotation of the drill string relative to the hook. As is well known, atop drive system could alternatively be used.

In this example embodiment, the surface system further includes drillingfluid or mud 26 stored in a pit 27 formed at the well site. A pump 29delivers the drilling fluid 26 to the interior of the drill string 12via a port in the swivel 19, causing the drilling fluid to flowdownwardly through the drill string 12 as indicated by the directionalarrow 8. The drilling fluid exits the drill string 12 via ports in thedrill bit 105, and then circulates upwardly through the annulus regionbetween the outside of the drill string and the wall of the borehole, asindicated by the directional arrows 9. In this manner, the drillingfluid lubricates the drill bit 105 and carries formation cuttings up tothe surface as it is returned to the pit 27 for recirculation.

The BHA 100 of the illustrated embodiment includes alogging-while-drilling (LWD) module 120, a measuring-while-drilling(MWD) module 130, a rotary-steerable system and motor 150, and drill bit105. The LWD module 120 may be housed in a special type of drill collar,as is known in the art, and can contain one or more types of loggingtools. It will also be understood that more than one LWD and/or MWDmodule can be employed, as represented at 120A. The LWD module includescapabilities for measuring, processing, and storing information, as wellas for communicating with the surface equipment. In the presentembodiment, the LWD module 120 and/or 120A may include a multiaxial(e.g., triaxial) electromagnetic logging tool.

The MWD module 130 is likewise housed in a special type of drill collar,as is known in the art, and can contain one or more devices formeasuring characteristics of the drill string and drill bit. The MWDtool further includes an apparatus (not shown) for generating electricalpower to the downhole system. This may typically include a mud turbinegenerator powered by the flow of the drilling fluid 26, although otherpower and/or battery systems may also be employed. By way of exampleonly, the MWD module 130 may include one or more of the following typesof measuring devices: a weight-on-bit measuring device, a torquemeasuring device, a vibration measuring device, a shock measuringdevice, a stick/slip measuring device, a direction measuring device, andan inclination measuring device. The operation of the assembly 10 ofFIG. 1B may be controlled using the logging and control system 152,which may include one or more processor-based computing systems. In thepresent context, a processor may include one or more microprocessors,such as one or more general-purpose microprocessors,application-specific microprocessors (ASICs), PLC, FPGA, SoC, or anyother suitable integrated circuit capable of executing encodedinstructions stored, for example, on tangible computer-readable media,or a combination of such processing components.

Instructions or data to be processed by the processor may be stored in acomputer-readable medium, such as a memory device, which may includevolatile memory, such as random access memory (RAM), non-volatilememory, such as read-only memory (ROM), or as a combination of RAM andROM devices. Such memory may store a variety of information and may beused for various purposes, including data acquisition routines, logginginversion algorithms, modeling functions, graphical user interfaces, aswell as any other suitable program, applications, or any other routinesthat may be executed by the well site systems in FIGS. 1A and 1B. Thecontrol system 152 may also include non-volatile storage for persistentstorage of data and/or instructions. For instance, the non-volatilestorage may include flash memory, a hard drive, or any other optical,magnetic, and/or solid-state storage media, or some combination thereof.Further, the control system 152 may include a network device thatenables the control system 152 connect to a network, such as a localarea network, a wireless network, or a cellular data network, and tocommunicate with other systems and devices over the network.

As will be explained in more detail below, the present disclosurerelates to an inversion technique that provides for handling varying dipand azimuth within the framework of a 1D inversion technique.Illustrative embodiments of the inversion technique disclosed herein maybe based on Taylor's expansion, which takes into account the effect ofvariation in dip and azimuth on the inversion. The application of thisTaylor series expansion approximation in an inversion may be implementedin a relatively straightforward manner.

An inversion process, in accordance with one embodiment of thedisclosure, may include a standard 1D inversion in which dip and azimuthare assumed constant over the inversion window at every iteration step,but with the addition of an additional iteration step in which dip andazimuth are allowed to change from bed to bed in the same inversionzone. In another embodiment, the dip azimuth is determined first beforea standard 1D inversion, and is precluded from both the standard 1Dinversion and the additional iteration step. In other words, dip azimuthis computed from raw data of the same tool directly or from other tool,and then fed to the forward solver of the inversion as a knownparameter.

Generally, a standard 1D inversion includes at least 10 iterations.Therefore, the increase in processing bandwidth using the presentlydisclosed inversion techniques is generally insignificant, since onlyone additional iteration step is needed. Thus, assuming a 10-iterationstandard 1D inversion, the increase in processing bandwidth and/or timeby adding the additional iteration step may be approximately 10-20percent. Additionally, in some implementations of the presentlydisclosed inversion techniques, it has been found that the increase inmemory requirements is approximately 30 percent or less, with theincrease in memory usage being caused primarily by the Jacobian matrix(for first order partial derivative), often the leading factor formemory consumption in 1D inversion methods. Overall, it will beappreciated by those skilled in the art that the inversion techniquesdescribed herein may be adapted for use with any electromagnetic loggingtool that provides measurements sensitive to formation dip and/or dipazimuth.

As an example only, an electromagnetic logging tool used in conjunctionwith the well systems shown in FIGS. 1A and 1B may include a triaxialelectromagnetic induction tool that has collocated triaxial transmitter(represented by T_(x), T_(y), T_(z)) and receiver coils (represented byR_(x), R_(y), R_(z)) to provide formation dip and dip azimuthinformation. FIG. 2A shows one possible arrangement of a transmitter anda receiver on such a triaxial measurement tool. The typical receiverwill include a main receiver (R_(x), R_(y), R_(z)) and a balancing or“bucking” receiver (B_(x), B_(y), B_(z)) that is used to cancel effectsof direct induction between the transmitter and the main receiver. Byway of example, an electromagnetic logging tool havingtransmitter-receiver configuration like that shown in FIG. 2 may be amodel the RT Scanner® tool, a triaxial induction logging tool availablefrom Schlumberger Technology Corporation of Sugar Land, Tex.

Measuring formations properties of a formation from within a wellboreusing a triaxial electromagnetic logging tools generally includesmeasuring nine component apparent conductivity tensors (σ_(a)(i,j,k),j,k=1, 2, 3), at multiple distances between an electromagnetictransmitter and the respective receivers, represented by index i. FIG.2B illustrates a vector the nine component apparent conductivity tensorfor one distance (spacing) that may be acquired using thetransmitter/receiver arrangement shown in FIG. 2A. FIG. 2C shows anotherembodiment of an electromagnetic logging tool design that incorporatestransverse transmitter (T_(xy)) and titled receivers (R_(zy,1),R_(xz,1), R_(yz,2), R_(xz,2)). It will be understood that any embodimentof the electromagnetic tool designs incorporating off-borehole axistransmitter and/or receiver coils, and any embodiment of theelectromagnetic tool designs incorporating only co-borehole axistransmitter and receiver coils (e.g., in the axial, or z-direction) buttraversing a formation at a high relative dip, are able to providemeasurements sensitive to formation dip and/or dip azimuth.

The measurements are usually obtained in the frequency domain byoperating the transmitter with a continuous wave (CW) of one or morediscrete frequencies to enhance the signal-to-noise ratio. However,measurements of the same information content could also be obtained andused from time domain signals through a Fourier decomposition process.This is a well known physics principle of frequency-time duality.

Formation properties, such as horizontal and vertical conductivities(Rh, Rv), relative dip angle (θ) and the dip azimuthal direction (Φ), aswell as borehole/tool properties, such as mud conductivity (σmud), holediameter (hd), tool eccentering distance (decc), tool eccenteringazimuthal angle (Ψ), can all affect these conductivity tensors. Theeffects of the borehole/tool to the measured conductivity tensors may bevery large even in an oil base mud (OBM) environment. Through aninversion technique, the above borehole/formation parameters may becalculated and the borehole effects can be removed from the measuredconductivity tensor. Some examples of techniques for removing boreholeeffects are discussed in commonly assigned International PatentApplication Publication No. WO2013/036896, hereby incorporated byreference in its entirety.

As discussed above, the inversion techniques described herein areapplicable to the measurements acquired with such tool designs toprovide formation resistivity (Rh and Rv), dip (θ) and/or dip azimuth(Φ) in the framework of a 1D inversion scheme, but with fewer 3D effectsand improved accuracy. Specifically, embodiments of the presenttechnique, which are particularly well suited for complex formationswith slowly varying dip and/or dip azimuth, provide an approximateforward model using a standard 1D formation model. A standard 1Dinversion is thus performed on an inversion zone (or window) where dipand/or dip azimuth are assumed to be constant and, thereafter, anadditional inversion using a Taylor expansion is performed using theapproximate forward model, but allowing dip and/or dip azimuth to varywithin the same inversion zone. For the purposes of this disclosure andto better differentiate from standard 1D inversion algorithms, theinversion techniques described herein may be referred to as “extended 1Dinversions.”

Several examples on different types of Taylor series expansions will nowbe described to better explain the implementation of the extended 1Dinversion process. They may be used in the described extended 1Dinversion as an approximate forward model. The application of theseTaylor series expansions to an extended 1D inversion will then bedescribed thereafter. The examples include: (1) Complete Taylor'sExpansion in Both Dip and Azimuth up to First Order; (2) ApproximatedTaylor's Expansion in Both Dip and Azimuth up to First Order; (3)Taylor's Expansion in Both Dip and Azimuth up to Zeroth Order; (4)Completed Taylor's Expansion in Dip (but not Azimuth) up to First Order;(5) Approximated Taylor's Expansion in Dip (but not Azimuth) up to FirstOrder; and (6) Approximated Taylor's Expansion in Dip (but not Azimuth)up to Second Order. The use of “complete” herein when describing suchexpansions refers to expansions that include the contribution from dipand azimuth of every bed in the formation model. The use of“approximate” or “approximated” herein when describing such expansionsrefers to expansions that include the contribution from dip and azimuthof only a host bed, wherein the host bed is where the sensitivity centerof the measurement is located.

The examples described below are discussed with reference to a syntheticformation model with varying dip and azimuth that is derived from anactual field study. Referring to FIG. 3, the top panel of FIG. 3 depictsboth vertical resistivity (Rv—as curve 180) and horizontal resistivity(Rh—as curve 182) of the formation. The middle panel of FIG. 3 shows thedip (curve 188) and dip azimuth (curve 186) of the formation in the samedepth zone.

As can be seen, the dip and dip azimuth over the depth zone shown inFIG. 3 are varying, wherein the level of variation describes thecomplexity of the formation. Thus, the formation is obviously no longer1D (which assumes a constant dip and/or dip azimuth). In the presentexample, the background dip and dip azimuth of the example formation are55 degrees and 120 degrees, respectively. When the actual varying dipand dip azimuth are replaced with their background counterparts, theformation model degenerates to a 1D formation model.

Example 1 Completed Taylor Series Expansion in Both Dip andAzimuth—First Order

Keeping the above example formation in mind, an example of a first orderTaylor series expansion for both dip and dip azimuth is now described.That is, the measurement is expanded in terms of dip and dip azimuth upto the first order. In this example, the following parameters aredefined: σ_(α) designates any one of the 9 components of the apparentconductivity tensor (FIG. 2B) at a given depth in the inversion zone;α_(n) and β_(n) represent dip and dip azimuth, respectively, of bed i,i=1 . . . N, where N is the number of beds in the inversion zone; vectorα is a vector containing the dips of all beds; and β is a vectorcontaining the dip azimuths of all beds; and α₀ and β₀ represent thebackground dip and dip azimuth, respectively.

In this example, consider the first order Taylor's expansion of σ_(α) inboth dip and dip azimuth, which is given by:

$\begin{matrix}{{\sigma_{a}\left( {\alpha,\beta} \right)} \approx {{\sigma_{a}\left( {\alpha_{0},\beta_{0}} \right)} + {\sum\limits_{i = 1}^{N}\; {\frac{\partial{\sigma_{a}\left( {\alpha_{i,0},\beta_{i,0}} \right)}}{\partial\alpha_{i}}\left( {\alpha_{i} - \alpha_{0}} \right)}} + {\sum\limits_{i = 1}^{N}\; {\frac{\partial{\sigma_{a}\left( {\alpha_{i,0},\beta_{i,0}} \right)}}{\partial\beta_{i}}\left( {\beta_{i} - \beta_{0}} \right)}}}} & (1)\end{matrix}$

In the above expansion, α₀=(α_(k,0)), β₀=(β_(k,0)), where α_(k,0)=α₀,and β_(k,0)=β₀. It is understood that, α₀ and β₀ correspond to the 1Dbackground and, therefore, σ_(α)(α₀, β₀)=σ_(α) ^(1D)(α₀, β₀). Thederivatives of σ_(α) with respect to α_(i) and β_(i) may be found byfirst computing the derivatives of the magnetic field with respect toa_(i) and β_(i) in the bedding coordinate system with the followingequations:

$\begin{matrix}{\frac{\partial H_{n_{T}n_{R}}}{\partial\alpha_{i}} = {I_{m}{l\left( {\sigma_{v,i} - \sigma_{h,i}} \right)}{\int_{V_{i}}\ {{{\overset{\_}{r}}^{\prime}\left( {{E_{x,n_{R}}E_{z,n_{T}}} + {E_{x,n_{T}}E_{z,n_{R}}}} \right)}}}}} & (2) \\{\frac{\partial H_{n_{T}n_{R}}}{\partial\beta_{i}} = {I_{m}{l\left( {\sigma_{v,i} - \sigma_{h,i}} \right)}\sin \; \theta {\int_{V_{i}}\ {{{\overset{\_}{r}}^{\prime}\left( {{E_{y,n_{R}}E_{z,n_{T}}} + {E_{y,n_{T}}E_{z,n_{R}}}} \right)}}}}} & (3)\end{matrix}$

In the above, I_(m)l is the dipole moment of the source corresponding tothe transmitter coil, and the transmitter and receiver coils areoriented in the n_(T) and n_(R) directions, respectively. E_(n) _(T)=(E_(x,n) _(T) , E_(y,n) _(T) , E_(z,n) _(T) )^(t) and E_(n) _(R)=(E_(x,n) _(R) , E_(y,n) _(R) , E_(z,n) _(R) )^(t) are the electricfield due to a source with a unit dipole moment corresponding to thetransmitter coil and an adjoint electric field due to a source with aunit dipole moment corresponding to the receiver coil, respectively;σ_(h,i) and represent horizontal conductivity and vertical conductivityof bed i, respectively; θ is the bedding dip in the tool coordinates,which is identical to α₀, the background dip as aforementioned; V_(i)designates the support of bed i. Here, the bedding coordinate system maybe defined such that the z-axis is along the normal to the bedding planeof the formation; and x- and y-axes are within the bedding plane. Thetool coordinate system is defined such that the z-axis is along the toolaxis; and x- and y-axes are within the tool plane, or the planeperpendicular to the tool axis. This process is repeated for eachpossible combination of transmitter and receiver coils (and bucking coilwherever applicable) in order to find derivatives of all possiblecouplings of the magnetic field tensor. Once the derivatives of magneticfield H_(n) _(R) _(n) _(T) with respect to α_(i) and β_(i) are found,they may be calibrated straightforwardly to those of apparentconductivity σ_(α) with respect to α_(i) and β_(i).

Even though the method for computing derivatives of apparentconductivity σ_(α) with respect to α_(i) and β_(i) are presented in theabove, i.e., those given in Equations (2) and (3) and the associateddescription, it will be apparent to those skilled in the art thatvariations may be applied to the above method in other embodiments. Thevariations may include but are not limited to applying the above methodto formations which are not non-magnetic, i.e. where the magneticpermeability μ are not equal to vacuum magnetic permeability μ₀; tosources which cannot be treated as point dipole; to electric-typesources; and to other types of measurements, e.g., phase shift andattenuation, etc.

Example 2 Approximated Taylor Series Expansion in Both Dip andAzimuth—First Order

In another example embodiment, the measurements may be expanded in termsof dip and dip azimuth using Taylor expansion up to first order using anapproximation for the derivatives of σ_(α) with respect to dip and dipazimuth. This may be given by:

$\begin{matrix}{{\sigma_{a}\left( {\alpha,\beta} \right)} \approx {{\sigma_{a}\left( {\alpha_{0},\beta_{0}} \right)} + {\frac{\partial{\sigma_{a}\left( {\alpha_{n,0},\beta_{n,0}} \right)}}{\partial\alpha_{n}}\left( {\alpha_{n} - \alpha_{0}} \right)} + {\frac{\partial{\sigma_{a}\left( {\alpha_{n,0},\beta_{n,0}} \right)}}{\partial\beta_{n}}\left( {\beta_{n} - \beta_{0}} \right)}}} & (4)\end{matrix}$

It is noted that the above expansion is not a complete first orderexpansion anymore, in that only the contribution from what may bereferred to as the “host bed” (bed n), where the sensitivity center ofσ_(α) falls, is considered in this example. The contributions from bedsother than bed n are ignored. The sensitivity center of σ_(α) is a placein the formation where σ_(α) exhibits the highest sensitivity. In oneembodiment, the approximation of the σ_(α) derivative (first order inthis case) with respect to dip and dip azimuth may be found inaccordance with the following equations:

$\begin{matrix}{\frac{\partial{\sigma_{a}\left( {\alpha_{n,0},\beta_{n,0}} \right)}}{\partial\alpha_{n}} \approx \frac{\partial{\sigma_{a}^{1\; D}\left( {\alpha_{0},\beta_{0}} \right)}}{\partial\alpha}} & (5) \\{\frac{\partial{\sigma_{a}\left( {\alpha_{n,0},\beta_{n,0}} \right)}}{\partial\beta_{n}} \approx \frac{\partial{\sigma_{a}^{1\; D}\left( {\alpha_{0},\beta_{0}} \right)}}{\partial\beta}} & (6)\end{matrix}$

Essentially, the right hand sides of Equations (5) and (6) are obtainedwith the background 1D formation model. It was found that Equations (5)and (6) provide a good and suitable approximation as long as σ_(α) andσ_(α) ^(1D) are in bed n and the bed is sufficiently thick. As anexample, a bed thickness of 1 to 2 feet is probably considered too thin,although sufficient thickness can also depend in part on resistivitycontrast. If the bed is thick enough, the left hand side and right handside of each of Equations (5) and (6) will be equal at the center of thebed. Thus, substituting the right hand side terms of Equations (5) and(6) back into Equation (4) yields the following:

$\begin{matrix}{{\sigma_{a}\left( {\alpha,\beta} \right)} \approx {{\sigma_{a}^{1\; D}\left( {\alpha_{0},\beta_{0}} \right)} + {\frac{\partial{\sigma_{a}^{1\; D}\left( {\alpha_{0},\beta_{0}} \right)}}{\partial\alpha}\left( {\alpha_{n} - \alpha_{0}} \right)} + {\frac{\partial{\sigma_{a}^{1\; D}\left( {\alpha_{0},\beta_{0}} \right)}}{\partial\beta}\left( {\beta_{n} - \beta_{0}} \right)}}} & (7)\end{matrix}$

Note that this approximated embodiment differs from the first example(complete) where the derivatives were calculated in accordance withEquations (2) to (3).

Example 3 Approximated Taylor Series Expansion in Dip and Azimuth—ZeroOrder

This next example illustrates that the simplest Taylor series expansionof dip and dip azimuth can be a zero order expansion. That is, thesimplest approximate solution is the one in the background formation:

σ_(α)(α, β)≈σ_(α) ^(1D)(α₀, β₀)  (8)

This equation can be understood as the one invoked at the last step of astandard 1D inversion to evaluate the misfit between field and simulateddata. Specifically, σ₀ and β₀ are understood as the inverted dip andazimuth obtained for the whole inversion zone. When the true formationis not 1D, the above equation represents a coarse approximation in thatall the higher order terms (e.g., first order and higher) are ignored.Therefore, it is appropriate to refer to it as the zero or zero-th orderapproximation.

Example 4 Complete Taylor Series Expansion in Dip (WithoutAzimuth)—First Order

To provide a further example, in accordance with one embodiment of thepresent disclosure, the electromagnetic measurements are expanded interms of dip up to the first order approximation while dip azimuth isdetermined prior to the inversion. For instance, in one embodiment, thedip azimuth may be computed from measurement of electromagnetic toolsdirectly. As will be appreciated, some electromagnetic logging toolsprovide a full conductivity tensor measurement which can be used toextract dip azimuth directly from the tensor components rather than byan inversion. As an example only, certain models of Rt Scanner®,available from Schlumberger Technology Corporation, are capable ofproviding such measurements. Some techniques for extracting dip azimuthdirectly from tensor components have been disclosed in InternationalPatent Application No. WO2008/137987 (commonly owned by the assignee ofthe present disclosure), Wu et. al., “Determining Formation Dip From AFully Triaxial Induction Tool,” presented at SPWLA 51st Annual LoggingSymposium, Perth Australia, Jun. 19-23, 2010, Wang et al., “TriaxialInduction Applications In Difficult and Unconventional Formations,”Transactions of the 53^(rd) SPWLA Annual Logging Symposium, Paper G(2012), and Wu et al., “Efficient Hierarchical Processing andInterpretation of Triaxial Induction Data in Formations With ChangingDip,” SPE 135442, presented at SPE Annual Technical Conference andExhibition, Florence Italy, Sep. 19-22, 2010, all of which areincorporated herein by reference. These above-referenced publicationsgenerally discuss various techniques for obtaining dip azimuth prior toinversion or to find an initial guess or estimation of dip azimuth in a1D inversion.

For the purposes of this disclosure, the finding of dip azimuth directlyfrom tool measurements (rather than through an inversion) may bereferred to as a “direct solution of dip azimuth,” a “pre-computed dipazimuth” or the like. This direct solution of dip azimuth has been shownto provide a good estimation of dip azimuth in complex formations withvarying dip and azimuth. Further, in other embodiments, direct solutionsof dip azimuth may also be found from logging tools other thanelectromagnetic tools, such as imaging tools (e.g., micro-resistivityimaging tools, oil-base mud imaging tools), density tools, and so forth.By way of example only, such imaging tools may include a model of FMI®Fullbore Formation MicroImager or OBMI® Oil-Base MicroImager, bothavailable from Schlumberger Technology Corporation. By using apre-computed dip azimuth in inversion directly, an approximate solutionfor dip may be obtained using the following expression:

$\begin{matrix}{{\sigma_{a}\left( {\alpha,\beta} \right)} \approx {{\sigma_{a}^{1\; D}\left( {\alpha_{0},\beta_{m}} \right)} + {\sum\limits_{i = 1}^{N}{\frac{\partial{\sigma_{a}\left( {\alpha_{0},\beta_{m}} \right)}}{\partial\alpha_{i}}\left( {\alpha_{i} - \alpha_{0}} \right)}}}} & (9)\end{matrix}$

Here, the derivative of σ_(α) with respect to dip may be determined asdescribed above with respect to Equation (2). In the above expansion andhereafter, β_(m) is the dip azimuth at the place corresponding to thesensitivity center of σ_(α), which may be found based on data from theabove-described electromagnetic tool for example. It is understood thatthe reverse is also possible, i.e., if dip is known or pre-computed, onecan solve for azimuth using a similar expansion.

Example 5 Approximated Taylor Series Expansion in Dip (WithoutAzimuth)—First Order

In another example, the measurement σ_(α) is expanded in terms of dip upto first order while dip azimuth is found before inversion using theapproximation for the derivative of σ_(α) with respect to dip. Themethods with which the dip azimuth are found is described above, whereinthe approximation may be given by:

$\begin{matrix}{{\sigma_{a}\left( {\alpha,\beta} \right)} \approx {{\sigma_{a}^{1\; D}\left( {\alpha_{0},\beta_{m}} \right)} + {\frac{\partial{\sigma_{a}^{1\; D}\left( {\alpha_{0},\beta_{m}} \right)}}{\partial\alpha}\left( {\alpha_{n} - \alpha_{0}} \right)}}} & (10)\end{matrix}$

In the above, the contribution from bed n is retained while those fromall the other beds are neglected. In addition, the derivative of σ_(α)with respect to dip is replaced with the one from the standard 1Dinversion in the inverted 1D model. Using the approximations inEquations (9) and (10) means that the dip azimuth will not be solved forin the inversion anymore. The pre-computed dip azimuth may be used asthe solution to dip azimuth instead. Moreover, the dip azimuth foundthis way may be different from depth point to depth point, instead ofbeing a constant over the inversion zone. Such a point-by-point dipazimuth is similar to what is offered by some prior arts based upon aformation model in which resistivity is assumed constant in thedirection along borehole axis.

Example 6 Approximated Taylor Series Expansion in Dip (WithoutAzimuth)—Second Order

In another embodiment, the measurement may be expanded in terms of dipup to the second order while dip azimuth is again determined prior toinversion (pre-computed). Moving forward one more step from Equation(10), the second order approximation can be obtained as follows:

$\begin{matrix}{{\sigma_{a}\left( {\alpha,\beta} \right)} \approx {{\sigma_{a}^{1\; D}\left( {\alpha_{0},\beta_{n}} \right)} + {\frac{\partial{\sigma_{a}^{1\; D}\left( {\alpha_{0},\beta_{n}} \right)}}{\partial\alpha}\left( {\alpha_{n} - \alpha_{0}} \right)} + {\frac{1}{2}\frac{\partial^{2}{\sigma_{a}^{1D}\left( {\alpha_{0},\beta_{n}} \right)}}{\partial\alpha^{2}}\left( {\alpha_{n} - \alpha_{0}} \right)^{2}}}} & (11)\end{matrix}$

As can be seen, the approximation set forth in Equation (11) includesthe second derivative of dip. From a computational standpoint, findingthe second derivative in addition to the first derivative may increaseprocessing time (e.g., approximately double in some cases) for dipderivative computation.

The performance of different approximation examples presented above canbe evaluated by comparing them against a 3D formation model. Forexample, to evaluate performance of these approximation techniques, toolresponses for different transmitter-receiver array spacings (15, 21, 27,39, 54, and 72 inches) in the formation were simulated with a 3D finitedifference method. By way of example only, an electromagnetic loggingtool having spacings similar to these may be a model of theabove-mentioned RT Scanner® tool. This set of synthetic data isconsidered exact in the sense that there is no approximation invoked inthe computation except for the numerical error.

A 1D forward solver is then invoked to compute the tool response and itsderivatives with respect to dip and dip azimuth in the background 1Dformation. The tool response and the derivatives in the background 1Dformation are then substituted in the approximate 3D models in Equations(7), (10), and (11) respectively to find an approximate 3D response forall arrays. These comparisons are summarized in FIGS. 3 and 4.Specifically, the bottom panel 200 of FIG. 3 shows the relative errorpercentages (representing data misfit of the simulated data) between the1D formation model given in Equation (8) (zero order expansion for dipand dip azimuth) and the exact 3D formation model for the variousexample array spacings given above (from 15 to 72 inches). As will beappreciated, Equation (8) may be referred to an approximation of a 1Dformation model, since dip and azimuth are assumed constant throughoutthe entire model for the zero-th order case (e.g., higher order termsare ignored).

The top panel 202, middle panel 204, and bottom panel 206 of FIG. 4 showthe relative error between the approximate 3D models given in Equations(7), (10), and (11) and the exact 3D model for the six example arraysspacings, respectively. The graph 210 in FIG. 5 shows the reduction ofdata misfit with the three approximate 3D models (from Equations (7),(10), and (11)). As can be seen in FIG. 5, the last approximate modelgiven by Equation (11) provides the best accuracy in that it exhibits agreater reduction in data misfit than the other approximations shown ingraph 210. Further, it can also be seen that among the different arrays,the shorter the array spacing is, the better the approximation is. Forinstance, for any of the given approximations, the shortest array, i.e.the 15 inch array, exhibits the largest reduction in data misfit.

While five examples of Taylor series expansion-based approximations arepresented herein, i.e., those given in Equations (1), (7), (9), (10),and (11) (the zero order example in Equation (8) is not being consideredan “expansion”), it will be apparent to those skilled in the art that anumber of variations may be applied to the approximation of toolmeasurements without departing from the concept and scope of thisdisclosure. For example, in another embodiment, a possible variation isto expand the measurement in terms of Taylor series expansion up to thesecond order of both dip and dip azimuth of the bed where thesensitivity center of the measurement falls. It is noted that this isdifferent from Equation (11) where dip azimuth is pre-computed. This canbe expressed as follows:

$\begin{matrix}{{\sigma_{a}\left( {\alpha,\beta} \right)} \approx {{\sigma_{a}^{1D}\left( {\alpha_{0},\beta_{0}} \right)} + {\frac{\partial{\sigma_{a}^{1D}\left( {\alpha_{0},\beta_{0}} \right)}}{\partial\alpha}\left( {\alpha_{n} - \alpha_{0}} \right)} + {\frac{\partial{\sigma_{a}^{1D}\left( {\alpha_{0},\beta_{0}} \right)}}{\partial\beta}\left( {\beta_{n} - \beta_{0}} \right)} + {\frac{1}{2}\frac{\partial^{2}{\sigma_{a}^{1D}\left( {\alpha_{0},\beta_{0}} \right)}}{\partial\alpha^{2}}\left( {\alpha_{n} - \alpha_{0}} \right)^{2}} + {\frac{1}{2}\frac{\partial^{2}{\sigma_{a}^{1D}\left( {\alpha_{0},\beta_{0}} \right)}}{\partial\beta^{2}}\left( {\beta_{n} - \beta_{0}} \right)^{2}}}} & (12)\end{matrix}$

In a further embodiment, the second order cross partial derivative ofdip and dip azimuth in expansion (12) may be further included:

$\begin{matrix}{{\sigma_{a}\left( {\alpha,\beta} \right)} \approx {{\sigma_{a}^{1D}\left( {\alpha_{0},\beta_{0}} \right)} + {\frac{\partial{\sigma_{a}^{1D}\left( {\alpha_{0},\beta_{0}} \right)}}{\partial\alpha}\left( {\alpha_{n} - \alpha_{0}} \right)} + {\frac{\partial{\sigma_{a}^{1D}\left( {\alpha_{0},\beta_{0}} \right)}}{\partial\beta}\left( {\beta_{n} - \beta_{0}} \right)} + {\frac{1}{2}\frac{\partial^{2}{\sigma_{a}^{1D}\left( {\alpha_{0},\beta_{0}} \right)}}{\partial\alpha^{2}}\left( {\alpha_{n} - \alpha_{0}} \right)^{2}} + {\frac{1}{2}\frac{\partial^{2}{\sigma_{a}^{1D}\left( {\alpha_{0},\beta_{0}} \right)}}{\partial\beta^{2}}\left( {\beta_{n} - \beta_{0}} \right)^{2}} + {\frac{\partial^{2}{\sigma_{a}^{1D}\left( {\alpha_{0},\beta_{0}} \right)}}{{\partial\alpha}{\partial\beta}}\left( {\alpha_{n} - \alpha_{0}} \right)\left( {\beta_{n} - \beta_{0}} \right)}}} & (13)\end{matrix}$

As will be appreciated, many other variations are still possible. Forexample, in still a further embodiment, another variation is to expandthe measurement in term of dip up to the second order, but in terms ofdip azimuth to the first order. In yet another embodiment, themeasurement may be expanded in terms of dip to the first order, but interms of dip azimuth up to the second order. Moreover, it should beunderstood that the presently disclosed techniques are not limited touse of a Taylor series expansion for the approximation of toolmeasurements in the framework of a 1D formation model. Rather, othertypes of mathematical expansions may be developed within the frameworkof a 1D formation model.

Extended 1D Inversion to Complex Formations with Slowly Varying Dip andAzimuth

The present section will now discuss in more detail how the exampleapproximations given above may be applied to an extended 1D inversion.Using the Born approximation techniques as an analogy, it can beappreciated that just as with Born approximation techniques where one isable to determine higher-dimensional resistivity distributions using asimple background, the presently described Taylor series expansionapproximation techniques allow for acquisition of more complex formationimage using a 1D formation model. It is also noted that the Bornapproximation is a linear approximation. In certain embodiments, theapproximation techniques set forth in the present disclosure may belinear approximations as well, or may be quadratic approximations (e.g.,having at least one second order term) in some instances. It should beunderstood that the Taylor series expansion approximations given aboveare generally most effective when the deviation of anomaly from thebackground medium is small (as is generally also the case for Bornapproximations). In other words, when practically applied to real worldapplications, the approximation techniques disclosed herein may workbest when the dip and dip azimuth vary slowly along the boreholetrajectory.

The following sections will explain in more detail how the extended 1Dinversion process is performed.

Cost Function for the Standard 1D Inversion

In accordance with embodiments of the present disclosure, two passes ofinversion are run in an extended 1D inversion. In the first pass, theextended 1D inversion is identical to a standard 1D inversion in thesense that dip and dip azimuth are assumed constant over the inversionzone, and an iterative solver is employed for the solution. Therefore,the unknown parameters are Rh, Rv in each bed, positions of each bedboundary, and a constant dip and dip azimuth over the inversion zone. Inone embodiment, the cost function of the standard 1D inversion can beexpressed as follows:

$\begin{matrix}{{{C(x)} = {{\frac{1}{2}{{W_{D}\left\lbrack {{d(x)} - d^{OBS}} \right\rbrack}}_{2}^{2}} + {\frac{1}{2}\lambda_{R}{{W_{R}\left( {x - x_{R}} \right)}}_{2}^{2}} + {\frac{1}{2}\lambda_{S}{{W_{S}{Dx}}}_{2}^{2}} + {\frac{1}{2}\lambda_{B}{{W_{B}p}}_{2}^{2}} + {\frac{1}{2}\lambda {x}_{2}^{2}}}}\mspace{79mu} {x = \left( {\sigma_{h}^{t},\sigma_{v}^{t},z^{t},\alpha,\beta} \right)^{t}}\mspace{79mu} {x_{R} = \left( {\sigma_{h,R}^{t},\sigma_{v,R}^{t},z_{R}^{t},\alpha_{R},\beta_{R}} \right)^{t}}} & (14)\end{matrix}$

In the above Equation (14), the formation is assumed to include N beds,with the two end beds being semi-infinitely thick. Therefore, horizontaland vertical conductivity σ_(h) and σ_(v) are each an N-dimensionalvector, and bed boundary z is an (N−1)-dimensional vector. The formationdip α and dip azimuth β are constant over the length of the inversionzone. Vector x_(R) is a reference model which represents a reasonableestimation for x, the unknown model. The third term of Equation (14) isprovided to produce a stable solution by limiting the variation ofresistivity along the borehole trajectory, where the matrix D is adifference operator. The fourth term in Equation (14) is used to applypre-established physical constraints with the plus function p, examplesof which are given in Wang et al., “Triaxial Induction Applications InDifficult and Unconventional Formations,” Transactions of the 53^(rd)SPWLA Annual Logging Symposium, Paper G (2012), and Wu et al.,“Efficient Hierarchical Processing and Interpretation of TriaxialInduction Data in Formations With Changing Dip,” SPE 135442, presentedat SPE Annual Technical Conference and Exhibition, Florence Italy, Sep.19-22, 2010, both incorporated herein by reference. Further, thematrices W_(D), W_(R), W_(S), and W_(B) each represent the weightingcoefficients for the components of the corresponding term. The finalterm in Equation (14) is used as a minimum length stabilizer, and thecoefficients λ_(R), λ_(S), λ_(B) and λ used in Equation (14) representthe regularization coefficients of the corresponding terms.

It will be appreciated that the implementation of this pass of theinversion is the same as most standard 1D inversion algorithms. Forexample, the formation is first segmented into a number of zones, andthe inversion is performed in the resultant subzones sequentially. Eachsubzone is referred to as an inversion zone. In the inversion of eachinversion zone, dip and dip azimuth are assumed constant over theinversion zone (sometimes also called inversion window). The length ofinversion zone is typically pre-selected before the inversion. Asexamples only, commonly used lengths for inversions zones may be in arange of approximately 20 to 100 feet. In one embodiment, an inversionwindow may be set to 30 feet, and further subdivided into upper, middle,and lower 10 feet sections. Using this example, only the results frommiddle section of the inversion may be output to reduce the truncationeffect.

As discussed above in discussing the various Taylor series expansionapproximation examples, the dip azimuth may be pre-computed in somecases (e.g., can be determined or otherwise derived directly from toolmeasurements without requiring an inversion). In such cases, dip azimuthwill be precluded from inversion as aforementioned. However, if this isthe case, it should be understood that the inversion is not necessarilybased on a true 1D model, since dip azimuth may not be constant in theinversion zone. In such cases, the inversion is sometimes referred to asa “pseudo 1D inversion.”

Cost Function at the Extended Step of the Extended 1D Inversion

After the standard 1D inversion in the first pass (discussed in thepreceding section) is finished, a second pass is performed. In thissecond pass, the restriction on the variation of dip and dip azimuthwithin an inversion zone is lifted so that dip and dip azimuth can varyfrom bed to bed. For example, in one embodiment, the cost function fromEquation (14) above is modified accordingly to accommodate this changeas follows:

$\begin{matrix}{{{{C(x)} = {{\frac{1}{2}{{W_{D}\left\lbrack {{d(x)} - d^{OBS}} \right\rbrack}}_{2}^{2}} + {\frac{1}{2}\lambda_{R}{{W_{R}\left( {x - x_{R}} \right)}}_{2}^{2}} + {\frac{1}{2}\lambda_{S}{{W_{S}{Dx}}}_{2}^{2}} + {\frac{1}{2}\lambda_{B}{{W_{B}p}}_{2}^{2}} + {\frac{1}{2}\lambda {x}_{2}^{2}}}}\mspace{79mu} {x = \left( {\sigma_{h}^{t},\sigma_{v}^{t},z^{t},\alpha^{t},\beta^{t}} \right)^{t}}}\mspace{79mu} {x_{R} = \left( {\sigma_{h,R}^{t},\sigma_{v,R}^{t},z_{R}^{t},\alpha_{R}^{t},\beta_{R}^{t}} \right)^{t}}} & (15)\end{matrix}$

When compared to the cost function of the first pass in Equation (14),the difference is that α, β, α_(R) and β_(R) are now each anN-dimensional vector, to be consistent with the aforementionedassumption (e.g., that dip and dip azimuth can vary within an inversionzone and from bed to bed). The matrices W_(R) and W_(B) can each beunderstood as being an updated form of these matrices resulting from thechange in dip and dip azimuth. Further, in another embodiment, the dipazimuth can be pre-computed, and β and β_(R) will be precluded from theabove formulation and hence from the inversion.

Referring now back to the first pass of the standard 1D inversion, theJacobian matrix can be given as follows:

J=(J_(h), J_(v), J_(z), J_(α), J_(β))  (16)

where J_(η), η=h,v is an M×N matrix, J_(η)=(δσ_(α,m)/δσ_(η,n)); J_(z) isan M×(N−1) matrix, J_(z)=(δσ_(α,m)/δz_(n)). Moreover, J_(α) and J_(β)are given as:

$\begin{matrix}{J_{\alpha} = \begin{pmatrix}\frac{\partial\sigma_{a,1}}{\partial\alpha} & \frac{\partial\sigma_{a,2}}{\partial\alpha} & \ldots & \frac{\partial\sigma_{a,M}}{\partial\alpha}\end{pmatrix}^{t}} & (17) \\{J_{\beta} = \begin{pmatrix}\frac{\partial\sigma_{a,1}}{\partial\beta} & \frac{\partial\sigma_{a,2}}{\partial\beta} & \ldots & \frac{\partial\sigma_{a,M}}{\partial\beta}\end{pmatrix}^{t}} & (18)\end{matrix}$

In one embodiment, the derivatives with respect to bed boundaries (orJacobian matrix J_(z)) are excluded from the inversion if bed boundariesare already obtained from other data. The same is true with thederivatives with respect to azimuth (or Jacobian matrix J_(β)) when theazimuth is directly computed from measurement data.

For the extended step in the second pass of the inversion, J_(h), J_(v)and J_(z) remain the same, but J_(α) and J_(β) are now each a fullmatrix, like J_(h) and J_(v). As discussed above, in one embodiment,J_(α) and J_(β) may be found in accordance with Equations (2) and (3)above. In another embodiment, the full matrices for J_(α) and J_(β) maybe approximated via the following approximation to obtain J_(α) andJ_(β):

$\begin{matrix}{{J_{x} = {\begin{pmatrix}\frac{\partial\sigma_{a,1}}{\partial x_{1}} & \frac{\partial\sigma_{a,1}}{\partial x_{2}} & \ldots & \frac{\partial\sigma_{a,1}}{\partial x_{N}} \\\frac{\partial\sigma_{a,2}}{\partial x_{1}} & \frac{\partial\sigma_{a,2}}{\partial x_{2}} & \ldots & \frac{\partial\sigma_{a,2}}{\partial x_{N}} \\\vdots & \vdots & \ddots & \vdots \\\frac{\partial\sigma_{a,M}}{\partial x_{1}} & \frac{\partial\sigma_{a,M}}{\partial x_{2}} & \ldots & \frac{\partial\sigma_{a,M}}{\partial x_{N}}\end{pmatrix} \approx \begin{pmatrix}J_{x_{1}} & J_{x_{2}} & \ldots & J_{x_{N}}\end{pmatrix}}},\mspace{79mu} {x = \alpha},\beta} & (19)\end{matrix}$

wherein:

$\begin{matrix}{{J_{x_{n}} = \begin{pmatrix}0 & \ldots & 0 & \frac{\partial\sigma_{a,m_{n,1}}}{\partial x_{n}} & \frac{\partial\sigma_{a,{m_{n,1} + 1}}}{\partial x_{n}} & \ldots & \frac{\partial\sigma_{a,m_{n,2}}}{\partial x_{n}} & 0 & \ldots & 0\end{pmatrix}^{t}},} & (20) \\{\mspace{85mu} {{x = \alpha},\beta}} & \;\end{matrix}$

In Equation (20), σ_(α,l), l=m_(n,1), . . . , m_(n,2), are themeasurements which fall in bed n. The rationale behind thisapproximation is that the measurements in bed n are most sensitive tothe dip and dip azimuth in the same bed. Furthermore, for the problem tobe tractable within the framework of 1D formation model, theapproximations in Equations (5) and (6) given above are employed:

$\begin{matrix}{{\frac{\partial\sigma_{a,l}}{\partial\alpha_{n}} \approx \frac{\partial\sigma_{a,l}^{1D}}{\partial\alpha}},} & (21) \\{\frac{\partial\sigma_{a,l}}{\partial\beta_{n}} \approx {\frac{\partial\sigma_{a,l}^{1D}}{\partial\beta}.}} & (22)\end{matrix}$

In other words, the derivatives with respect to dip and dip azimuth overthe inversion zone may be used to approximate the derivatives withrespect to those in bed n. The rationale for this approximation is thesame as that discussed for Equation (20), i.e., that measurements in bedn are most sensitive to dip and dip azimuth in the same bed. Bothapproximations are based on the localized property of electromagneticmeasurements. Particularly, δσ_(α,l)/δα_(n) corresponds to the change ofσ_(α,l) with respect to the change in the dip of bed n, whereas δσ_(α,l)^(1D)/δα corresponds to the change with respect to the change of dip inthe whole inversion zone. It can be seen that these expressions are notidentical. However, since σ_(α,l) is in bed n, it is most sensitive tochanges in dip within this bed. Therefore, the change of dip in bed nwill typically dominate over the change of dip in all other beds of theformation model. This observation thus lays the foundation for Equations(21) and (22). In theory, the accuracy of Equations (21) and (22) ishigher for measurements at the center of a thick bed. For this reason,it is sometimes useful to use a thick segmentation in the inversionprocess. In addition, in the extended step of the inversion, droppingthe measurements closer to bed boundaries is expected to give generallya better dip estimation. Further, as mentioned above, in certainembodiments, the dip azimuth can be computed directly without inversion.In such cases, the Jacobian matrix J_(β) will not be invoked in theinversion and the approximation in Equation (20) regarding dip azimuth(x=β) and that in Equation (22) will be unnecessary.

While the approximation of the Jacobian matrix has been described withrespect to a specific embodiment above, those skilled in the art willappreciate that variations may be applied to the approximation. Forexample, in one embodiment, the approximation is made on the Jacobianmatrices of Rh and Rv in a way similar to that used for J_(α) and J_(β),with derivatives with respect to Rh and Rv being set to zero in the bedsother than the host bed (bed n) where the sensitivity center of themeasurement falls. In so doing, J_(h) and J_(v) will be of the samestructure as J_(α) and J_(β). In other embodiments, a quadraticexpansion (e.g., having at least one second order term) can be used, inwhich the measurements in terms of dip and/or dip azimuth are expandedto the second order (including but not limited to the examples shownabove in Equations (11), (12), and (13)). Quadratic expansions of thistype can be invoked to further improve the approximation of the Hessianmatrix (e.g., a square matrix of second-order partial derivatives) ofthe cost function. It will be appreciated by those skilled in the artthat the Hessian is the second order counterpart to the Jacobian, andthe improvement on the Hessian in this manner is accomplished byproviding the curvature information of the cost function (e.g., Equation(12)). See Habashy et al., “A General Framework for ConstraintMinimization for the Inversion of Electromagnetic Measurements,”Progress in Electromagnetic Research, vol. 46, pp. 265-312 (2004).

Again, it is also emphasized that the presently disclosed techniques arenot limited to use of a Taylor series expansion for approximation of theJacobian and Hessian matrices within the framework of a 1D formationmodel. Other suitable types of mathematical expansions may be developedwithin the framework of a 1D formation model.

FIGS. 6 and 7 provide flowcharts showing extended 1D inversion processesin accordance with embodiments of the present disclosure. For example,FIG. 6 shows the flowchart of one embodiment (600) in which both dip anddip azimuth are found via inversion. Specifically, dip and dip azimuthare first inverted in the first pass of a standard 1D inversion byassuming they are constant over the inversion zone (602-616). Then, in asubsequent second pass of the inversion, they are allowed to change frombed to bed in the inversion zone, such that upon output, they are each avector representing dip or dip azimuth in all beds (the same as thehorizontal and vertical resistivity components), instead of a scalarcorresponding to the whole inversion zone as in the first pass(618-628).

FIG. 7 shows the flowchart of another embodiment (700) of the presentdisclosure in which only dip (and not dip azimuth) is found viainversion. Specifically, dip azimuth can be pre-computed, such as bybeing calculated or otherwise determined directly from tool measurements(e.g., computed with some components of the conductivity tensor asaforementioned). In this case, the dip azimuth is entered into theinversion as a known parameter, with the unknown parameters beinghorizontal and vertical resistivity, and formation dip. The dip is firstinverted in the first pass of a standard 1D inversion by assuming it isconstant over the inversion zone (702-718). Then, in the second pass ofthe inversion, dip is allowed to change from bed to bed in the inversionzone and, upon output, it returns a vector consisting of the dip of allbeds (the same as horizontal and vertical resistivity vectors), insteadof a scalar corresponding to the whole inversion zone as is the case inthe first pass of the inversion (720-730).

Application of the Extended 1D Inversion on Synthetic and Field Examples

This section of the disclosure is intended to provide some additionalexamples showing the application of the extended 1D inversion techniqueapplied to both synthetic and field examples. In these examples, datafrom an orthogonally collocated multi-array induction tool was used todemonstrate the performance of the extended 1D inversion technique. Themulti-array took may have a number of array spacings (e.g., 15, 21, 27,39, 54, and 72 inches). One array is shown schematically in FIG. 2A. Inthe following examples, it can be assumed the azimuth is found frommeasurement data directly (pre-computed), so it will not be an unknownparameter in the inversion. In addition, for the examples discussedbelow, it can also be assumed that an inversion window of 30 feet isused, with the window further subdivided into upper, middle, and lower10 feet sections. Using this example, only the results from middlesection of the inversion may be output to reduce truncation effects.

With this in mind, FIG. 8 shows the extended 1D inversion in a 1Dformation with constant dip and dip azimuth. The formation model isextracted from a field log as shown by the blue and red square logs ofRh (250) and Rv (252) in the upper panel. The true dip and dip azimuthare 55 degrees and 120 degrees respectively as shown by the blue (256)and red (254) straight lines in the lower panel of FIG. 8. The invertedRh, Rv, and dip with a 72 inch array of an electromagnetic logging toolare shown in green and blue square logs (inverted Rh curve 260, invertedRv curve 262) in the top panel, with inverted dip (circles 266) shown inthe lower panel of FIG. 8. Though dip azimuth is labeled as “inverted”,for this example, the dip azimuth is actually pre-computed from the rawdata directly (represented by circles 268 in lower panel of FIG. 8). Theresults shown here indicate that as an inversion designed for complexformations with slowly varying dip and dip azimuth, the extended 1Dinversion is able to provide fairly good estimations of Rh, Rv, dip anddip azimuth in 1D formations as a standard 1D inversion. It is notedthat the approximation of J_(α), as given in Equations (20) and (21) maybe employed in the extended inversion with respect to the followingsynthetic and field examples.

FIG. 9 shows the inversion results as obtained with the extended 1Dinversion in a 3D formation. In comparison to the above 1D formation,the difference is that now dip and dip azimuth are varying along theborehole trajectory over the whole depth zone. They are plotted assquare logs (curve 270 for dip azimuth, curve 272 for dip) in the lowerpanel of FIG. 9. As can be seen, the dip varies from approximately 37degrees to 70 degrees, and the dip azimuth can be varies fromapproximately 100 degrees to 135 degrees. The variation in dip isrelatively larger than that in dip azimuth in this case. Formations withsuch level of variation in dip and dip azimuth are not uncommon in fieldcases. The Rh and Rv square logs (curves 250 and 252, respectively inFIG. 9) are the same as those from the FIG. 8 example and thus have beenlabeled with the same reference numbers.

The inverted Rh, Rv, and dip with the 72 inch array of theaforementioned example electromagnetic logging tool are shown in squarelogs (curve 280 for inverted Rv, curve 282 for inverted Rh), and circles286 for inverted dip. The directly computed dip azimuth is also plottedin the lower panel of FIG. 9. Again, though labeled as “inverted”, forthis example, the dip azimuth is actually pre-computed from the rawdata, such as directly from electromagnetic tool measurements(represented by circles 288 in lower panel of FIG. 9). These resultsshow that in a complex formation (e.g., 3D), the extended 1D inversioncan still provide a reasonable estimation of Rh, Rv, dip and dipazimuth. In particular, the dip estimation is greatly improved.

For the sake of comparison, we switched off the second pass of theextended 1D inversion process, and re-ran the inversion. Thecorresponding results are plotted in FIG. 10. The square logs for Rh andRv and for dip and dip azimuth are the same as those in FIGS. 8 and 9.However, as can be seen, the inverted dip (circles 295) is found to beconstant over every 10 feet zone in the illustrated depth range. This 10feet extent corresponds to the size of the middle part of the 30 feetinversion zone. Normally only the results in the middle part of theinversion zone are kept and output to reduce the truncation effect.Thus, in comparison to the dip estimate with the regular extended 1Dinversion of FIG. 9, the results in FIG. 10 show that when the secondpass is omitted, the dip result is essentially a low pass filtered truedip in which the high frequency components are lost.

FIG. 11 shows Rh, Rv, dip and dip azimuth obtained with the extended 1Dinversion in a field WBM (water-base mud) case. Rh and Rv are shown inthe first track, and dip and dip azimuth in the second track areobtained with a standard 1D inversion. Rh and Rv in the third track, anddip and dip azimuth in the fourth track are obtained with the extended1D inversion. The dip and dip azimuth in the right-most track areobtained with a 0D inversion. Note that the horizontal resistivitymildly changes over the entire zone. This represents a favorablesituation for a 0D inversion. The dip with the 0D inversion may serve asa good reference for the true formation dip. Indeed, it can be seen thatRh and Rv with the two 1D inversions are very similar. In contrast, thedip with the extended 1D inversion follows the 0D dip more closely thanthe standard 1D inversion.

In a further aspect, formation complexity may be determined using dipand/or dip azimuth obtained from the extended 1D inversion by findingthe variation of dip and/or azimuth with respect to measured depth ortrue vertical depth. Further, as is well understood, instead of dipazimuth, strike (e.g., intersection between horizontal plane and beddingplane) may also be used to characterize formation orientation inconjunction with dip. The varying strike of a complex formation can bedetermined with the same techniques as that for dip azimuth as describedin the present disclosure.

As will be understood, the various techniques described above andrelating to extended 1D inversions are provided herein by way of exampleonly. Accordingly, it should be understood that the present disclosureshould not be construed as being limited to only the examples providedabove. Further, it should be appreciated that the extended 1D inversiontechniques disclosed herein may be implemented in any suitable manner,including hardware (suitably configured circuitry), software (e.g., viaa computer program including executable code stored on one or moretangible computer readable medium), or via using a combination of bothhardware and software elements.

While the specific embodiments described above have been shown by way ofexample, it will be appreciated that many modifications and otherembodiments will come to the mind of one skilled in the art having thebenefit of the teachings presented in the foregoing description and theassociated drawings. Accordingly, it is understood that variousmodifications and embodiments are intended to be included within thescope of the appended claims.

What is claimed is:
 1. A method to determine at least one formationproperty of a subterranean formation, comprising: providing a downholeelectromagnetic logging tool having at least one transmitter array andone receiver array; acquiring measurements in the formation using thetransmitter and receiver arrays of the downhole electromagnetic loggingtool; performing a first inversion in response to the measurements,wherein at least one of dip or dip azimuth are assumed constant in aninversion zone within the formation to obtain an inverted formationmodel that comprises at least one of horizontal resistivity (Rh),vertical resistivity (Rv), dip, and dip azimuth; determining an n-thorder partial derivative matrix of at least one of dip or dip azimuth,wherein n is greater than or equal to 1; performing a second inversionusing the determined n-th order partial derivative matrix, wherein atleast one of dip and dip azimuth are allowed to vary in the inversionzone, to obtain an updated formation model; and determining the at leastone formation property of the formation using the updated formationmodel.
 2. The method of claim 1, wherein n=1 and the partial derivativematrix comprises a Jacobian matrix of at least one of dip or dipazimuth.
 3. The method of claim 1, wherein determining the n-th orderpartial derivative matrix comprises using a mathematical expansion. 4.The method of claim 3, wherein the mathematical expansion comprisesusing an approximated expansion that ignores contribution of at leastone of dip or dip azimuth from beds in the inverted formation modelother than a host bed in which a sensitivity center of the measurementsis located.
 5. The method of claim 3, wherein the mathematical expansioncomprises using a complete expansion that takes into accountcontributions of at least one of dip or dip azimuth from each bed in theinverted formation model.
 6. The method of any of claims 1-5, whereinthe mathematical expansion comprises a Taylor series expansion.
 7. Themethod of claim 5, wherein the mathematical expansion is performedaccording to:${\sigma_{a}\left( {\alpha,\beta} \right)} \approx {{\sigma_{a}\left( {\alpha_{0},\beta_{0}} \right)} + {\sum\limits_{i = 1}^{N}{\frac{\partial{\sigma_{a}\left( {\alpha_{i,0},\beta_{i,0}} \right)}}{\partial\alpha_{i}}\left( {\alpha_{i} - \alpha_{0}} \right)}} + {\sum\limits_{i = 1}^{N}{\frac{\partial{\sigma_{a}\left( {\alpha_{i,0},\beta_{i,0}} \right)}}{\partial\beta_{i}}\left( {\beta_{i} - \beta_{0}} \right)}}}$wherein α₀=(α_(k,0)), β₀=(β_(k,0)), where α_(k,0)=α₀, and β_(k,0)=β₀,wherein α₀ and β₀ represents the dip and dip azimuth, respectively, fromthe inverted formation model, and wherein the n-th order partialderivative matrix is determined as a first-order partial derivativematrix that includes first-order partial derivatives for dip and dipazimuth determined using:$\frac{\partial H_{n_{T}n_{R}}}{\partial\alpha_{i}} = {I_{m}{l\left( {\sigma_{v,i} - \sigma_{h,i}} \right)}{\int_{V_{i}}{{{\overset{\_}{r}}^{\prime}\left( {{E_{x,n_{R}}E_{z,n_{T}}} + {E_{x,n_{T}}E_{z,n_{R}}}} \right)}}}}$$\frac{\partial H_{n_{T}n_{R}}}{\partial\beta_{i}} = {I_{m}{l\left( {\sigma_{v,i} - \sigma_{h,i}} \right)}\sin \; \theta {\int_{V_{i}}{{{\overset{\_}{r}}^{\prime}\left( {{E_{y,n_{R}}E_{z,n_{T}}} + {E_{y,n_{T}}E_{z,n_{R}}}} \right)}}}}$wherein H_(n) _(R) _(n) _(T) is a magnetic field at a receiver of thereceiver array oriented in the n_(R) direction in response to atransmitter of the transmitter array oriented in the n_(T) direction,I_(m)l is a dipole moment of a source corresponding to the transmitter,E_(n) _(T) =(E_(x,n) _(T) , E_(y,n) _(T) , E_(z,n) _(T) )^(t) is anelectric field due to the source with a unit dipole moment correspondingto the transmitter, E_(n) _(R) =(E_(x,n) _(R) , E_(y,n) _(R) , E_(z,n)_(R) )^(t) is an electric field due to a source with a unit dipolemoment corresponding to the receiver, σ_(h,i) and σ_(v,i) representhorizontal conductivity and vertical conductivity of bed i,respectively, θ is a bedding dip in accordance with a tool coordinatesystem, V_(i) designates the support of bed i, and α_(i) and β_(i)represent dip and dip azimuth, respectively of bed i.
 8. The method ofclaim 1, comprising using at least one of dip or dip azimuth from theupdated formation model to find variation in at least one of dip orazimuth with respect to measured depth or true vertical depth todetermine complexity of the formation.
 9. The method of claim 2, whereinn=2 and the partial derivative matrix comprises a Hessian matrix of atleast one of dip or dip azimuth.
 10. A system comprising: anelectromagnetic logging tool having at least one transmitter array andone receiver array, the electromagnetic logging tool being configured toacquire measurements in a subterranean formation using the transmitterand receiver arrays when disposed in the formation; a processing deviceconfigured to: perform a first inversion in response to the measurementswherein at least one of dip or dip azimuth are assumed constant in aninversion zone within the formation to obtain an inverted formationmodel that comprises at least one of horizontal resistivity (Rh),vertical resistivity (Rv), dip, and dip azimuth; determine an n-th orderpartial derivative matrix of at least one of dip or dip azimuth, whereinn is greater than or equal to 1; perform a second inversion using thedetermined n-th order partial derivative matrix, wherein at least one ofdip and dip azimuth are allowed to vary in the inversion zone, to obtainan updated formation model; and determine at least one formationproperty of the formation using the updated formation model.
 11. Thesystem of claim 10, wherein the electromagnetic logging tool is conveyedusing at least one of drill pipe, wireline, slickline, or coiled tubing.12. The system of claim 10, wherein the electromagnetic logging toolcomprises a wireline or an logging-while-drilling tool.
 13. The systemof claim 10, wherein n=1 and the partial derivative matrix comprises aJacobian matrix of at least one of dip or dip azimuth.
 14. The system ofclaim 10, wherein the processing device determines the n-th orderpartial derivative matrix comprises using a mathematical expansion. 15.The system of claim 14, wherein the mathematical expansion comprisesusing an approximated expansion that ignores contribution of at leastone of dip or dip azimuth from beds in the inverted formation modelother than a host bed in which a sensitivity center of the measurementsis located.
 16. The system of claim 14, wherein the mathematicalexpansion comprises using a complete expansion that takes into accountcontributions of at least one of dip or dip azimuth from each bed in theinverted formation model.
 17. The system of any of claims 10-16, whereinthe mathematical expansion comprises a Taylor series expansion.
 18. Anon-transitory computer-readable medium having computer executableinstructions for causing a computer to: in response to electromagneticmeasurement obtained by an electromagnetic logging tool disposed in aformation, perform a first inversion wherein at least one of dip or dipazimuth are assumed constant in an inversion zone within the formationto obtain an inverted formation model that comprises at least one ofhorizontal resistivity (Rh), vertical resistivity (Rv), dip, and dipazimuth; determine an n-th order partial derivative matrix of at leastone of dip or dip azimuth, wherein n is greater than or equal to 1;perform a second inversion using the determined n-th order partialderivative matrix, wherein at least one of dip and dip azimuth areallowed to vary in the inversion zone, to obtain an updated formationmodel; and determine at least one formation property of the formationusing the updated formation model.
 19. The non-transitorycomputer-readable medium of claim 18, wherein the determination of then-th order partial derivative matrix comprises using a mathematicalexpansion.
 20. The non-transitory computer-readable medium of claim 19,wherein the mathematical expansion comprises using an approximatedexpansion that ignores contribution of at least one of dip or dipazimuth from beds in the inverted formation model other than a host bedin which a sensitivity center of the measurements is located.
 21. Thenon-transitory computer-readable medium of claim 19, wherein themathematical expansion comprises using a complete expansion that takesinto account contributions of at least one of dip or dip azimuth fromeach bed in the inverted formation model.